Peter:After you draw circles F and H, you have FD=FN and HD=HN. Then FH is the perpendicular bisector of DN. [A rect if perpendicular bisector of a segment if two points of that rect are equal distanced from the segment extremes]

In the 1st part of my solution, by drawing circles F and H we have FD=FN' and HD=HN'( N and N' are 2 separated points)Note that N is the intersecting point of DM to circumcircle of triangle ABC while N' is the intersecting point of circle F to circle H . I try to prove that these 2 points N and N' coincide . If these 2 points coincide then FH become perpendicular bisector of DN . Hope that It will clear any confusion .Peter Tran